Remember Kelly and her pre-trip meeting? Have you ever been hiking on difficult terrain?

Well, Kelly figured out how many miles per hour her group would be able to hike on difficult terrain. However, she also had other thoughts about flat terrain. Look at this information.

“The amount of distance traveled depends on the group,” said Scott one of the leaders. “If your group works well together, they you can cover about 1 – 2 miles per hour on flat terrain. If you are climbing higher ascents, then you may only cover \begin{align*}\frac{1}{2}\end{align*} miles to 2 miles per hour.”

This is what the group leader told Kelly and the other hikers. Kelly has figured out the difficult hiking and now she has another idea of a problem to solve. Kelly wants to figure out the number of miles possible between the range of \begin{align*}\frac{1}{2}\end{align*} mile per hour and 2 miles per hour. She is sure that there is a way to do it using a variable and the six hours that the group will hike.

Pay close attention during this Concept and you will know how to use a variable expression to help Kelly figure things out.

Guidance

In the last Concept, you learned how to evaluate algebraic expressions with a single variable. Well, algebraic expressions can have more than one variable. Look at the following situations with multi-variable expressions.

When we know the value of the variables, we can evaluate multi-variable expressions the same way we evaluated single-variable expressions, by substituting the value for the variables in the expression and solving from left to right.

In this case we are only given one possible value for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. We know that \begin{align*}x = 2\end{align*} and \begin{align*}y = 4\end{align*}, we can evaluate the expression using the given values.

First, we can rewrite the expression by substituting the given values into the equation.

\begin{align*}& xy + x\\
& (2)(4) + 2\end{align*}

We used the parentheses here to show multiplication. When two variables are next to each other it means multiplication. Here we used the parentheses because we needed to show multiplication between 2 and 4.

Now back to Kelly. Once she has figured out that the group will probably go three miles per hour on difficult terrain, she moved on to other terrains.

Next, Kelly looks at medium terrain. She substitutes 1 mile into the expression for \begin{align*}x\end{align*}.

\begin{align*}6(1)=6\end{align*}miles on medium terrain

Next, Kelly looks at flat terrain. She substitutes 2 miles in for \begin{align*}x\end{align*}.

\begin{align*}6(2) = 12\end{align*}miles on flat terrain

Kelly starts to think about this. It would be unlikely for the group to travel on all one type of terrain per day. So she writes this expression to show half the day on flat terrain and half the day on difficult terrain.

Kelly looks at this figure. While she is estimating travel time and distance per miles, she figures that the group may cover a little less than 7 and one-half miles or a little more, but it is probably a good middle estimate for distance covered per day.